Some Basic Properties of Integers --------------------------------- The Well-Ordering Principle --------------------------- Any nonempty set of integers bounded from below has a least element. Similarly, any nonempty set of integers bounded from above has a greatest element. The elements referred to are unique. The Division Theorem -------------------- Let a, b be integers, b != 0. Then there exists unique integers q,r such that a = qb + r where 0<= r < b We gave a geometric argument for the Division theorem. Divisibility ------------ We defined the divisibility relation as follows: Let a, b be integers, a != 0. Then a | b if there exists a number q such that aq = b. We claimed the following basic properties of divisibility: reflexive: a|a transitive a|b and b|c -> a|c (the symmetric property does not hold: a|b does not imply that b|a) if d|a and d|b then d | ax+by for any x,y Primes and Composites --------------------- A number n>=2 is primes a|n implies a = 1, -1, n, or -n. A number is composite if it is not prime. GCD --- let a,b be integers, not both 0. Then gcd(a,b) = the largest number d such that d|a and d|b. Euclid's algorithm ------------------ Repeatedly use the following fact: if a = qb + r where 0<= r < b then gcd(a,b) = gcd(b,r) Your book gives examples (pp 24-25). The same algorithm lets you recover x, y such that gcd(a,b) = ax + by